Suppose I change the state of a
quantum system ``smoothly''. For example, I could move the system through
space, or I could ``move it through time'' (i.e., just wait - hence the
term, ``evolution operator''), or I could (surprise!) rotate it.

It happens quite generally in quantum mechanics that all such state changes
are induced by unitary operators. Proving this clearly would require some
physical assumptions, and I won't go into this at all. For our spinning
electron, it turns out you can even assume a little more: the change of
state caused by rotating the electron is induced by an operator in .
The operator in question is called a rotation operator.

How should we visualize the action of a rotation operator on a state
vector ? We saw how to picture as a rotation in 3-space by looking
at its effects on traceless Hermitian matrices:
, where
. How can we hook up the action of on
state vectors with the action of on traceless Hermitian matrices? It
seems we need a correspondence between states (say ) and matrices of
the form
. We won't get quite this, but
we'll get something just as good.

The trick is to set up a correspondence between states and yet another kind
of matrix: a projection matrix. You probably noticed that the matrix
does a pretty good job specifying the state ``spin up along the
z-axis''. As it turns out, is not a projection matrix, but it
corresponds in a natural fashion to
,
which is.

Here's how it goes for an arbitrary state vector
.
Suppose is normalized, so . The projection matrix for
is given by taking the product of the column vector with the row
vector :

(Standard physicists' notation is for and for
. The product is
. The norm is
. Mathematicians prefer to talk about a vector space and its
dual instead of column vectors and row vectors, but these notes prefer
concreteness to elegance.)

Now, is a Hermitian matrix with determinant 0 (check!). It must
therefore take the form:

(If the appearence of
makes you think ``Special
Relativity!'', you're on the right track, but I won't get into that.)
However, the trace is not 0, but . Since I took to be
normalized (), it follows that .

So we have a mapping from state vectors to Hermitian matrices of the form
with
. And the latter are in an obvious one-one
correspondence with points on a sphere of radius one-half.

The mapping (restricted to normalized vectors) actually
establishes a one-one correspondence between states and our special
class of Hermitian matrices. For let be a complex number of norm 1;
then
.

Why do I call a projection matrix? Answer: by analogy with
projections in ordinary real vector spaces, say . If
is a vector of norm 1, and is an arbitrary vector,
then the projection of ``along the vector ''
(i.e., in the subspace spanned by ) is . Analogous to this, we define
, using the notation
for the inner product. In the
``row vector, column vector'' notation, this is
.
In physicists' notation, this is
.

We have acquired a new way of picturing the action of on 3-space.
The formula
captures it succinctly. The mapping
sets up a one-one correspondence between the states
(i.e., the complex projective line) and points on a sphere in 3-space. In
fact this is just the Riemann sphere mapping!

So the quantum states for the spin of an electron can be pictured as points
on a sphere. Elements of correspond to the change in state induced
by rotating the electron, and this action of can be pictured as a
rotation of the sphere. The naive pictures match up with the
formalism flawlessly. The element of induces the identity
mapping on the space of states, since and represent the same
quantum state.

A simple computation illustrates how everything meshes. The rotation
operator for a clockwise rotation about the y-axis is
. Indeed, if you work out
, you get , and
likewise
. The x-axis
maps to the z-axis, and the z-axis maps to minus the x-axis.

The example of electron spin illustrates two features of quantum
mechanics very clearly.

Probabilistic character: The quantum state does not
uniquely determine the result of experiment. Hence
Einstein's famous complaint, ``I shall never believe that God plays
dice with the universe.'' (Perhaps he plays cards with the
physicists?)

Some more technical features, also embodied in this example, and typical of
quantum mechanics:

Need for complex numbers: The neat correspondence with the
classical spinning ball picture wouldn't work if we did everything over
R.

Non-commuting observables: You cannot simultaneously measure
the x and z components of spin (for example), because and
do not commute.

Symmetry groups and observables: The rotation symmetry group
gives rise indirectly to the matrices, and ultimately to the
notion of angular momentum. The mathematical basis is the Lie
groupLie algebra correspondence.

Had I started with the first historical example, the single spinless
particle coasting in space, I would be illustrating the same morals with
different actors:

Need for complex numbers: The appropriate Hilbert space is the
space of complex-valued functions on 3-space.

Non-commuting observables: The momentum and position operators
do not commute, and you cannot simultaneously measure position and
momentum.

Symmetry groups and observables: The group of translations in
3-space gives rise to Lie group acting on the Hilbert space; the
momentum operator emerges from the corresponding Lie algebra.